The home secretary, Theresa May, today promised a review of Britain's gun laws following yesterday's shootings in Cumbria, but said it would be wrong to act before the full facts about Derrick Bird's killing spree were known.

David Cameron later backed up her comments, warning against any rash response, an approach endorsed too by Labour's former home secretary Alan Johnson. The prime minister and May plan to visit the area tomorrow to make sure the police, emergency and health services and local councils have all the support they need.

May told MPs: "Undoubtedly, yesterday's killings will prompt a debate about our country's gun laws. That is understandable – and indeed it is right and proper. But it would be wrong to react before we know the full facts. Today we must remember the innocent people who were taken from us as they went about their lives. Then, we must allow the police time to complete their investigations."

Cameron, speaking at a press conference in Downing Street, said: "The right thing to do is, of course, to look at all of these issues and have an open mind.

"But we should be clear that in this country we have some of the toughest gun control legislation anywhere in the world and we shouldn't make any kneejerk reaction to think that there is some instant legislative or regulatory answer."

The prime minister added: "Of course we have to do everything we can to stop these dreadful events, but you can't legislate to stop a switch flicking in someone's head and for this dreadful sort of action to take place."

Cumbria police confirmed that Bird – who went on the rampage in the UK's most deadly shooting incident since the Dunblane massacre – had been a licensed gun holder. May, in a statement to the House of Commons, said he had held a shotgun licence since 1995 and a firearms licence (for a .22 rifle) since 2007, adding that police had confirmed to her that his licences covered the firearms seized yesterday.

Twelve people were killed and eight remained in hospital today after the 52-year-old taxi driver went on the apparently indiscriminate shooting spree.

Bird shot dead his twin brother and at least one colleague before driving through rural west Cumbria firing seemingly at random at people in towns, villages and on country roads before killing himself.

More than 100 detectives are beginning to piece together the sequence of events and trying to establish his motive.

Eleven people were injured during the three-and-a-half hour incident, which paralysed the county as police, hunting the gunman on the ground and by air, ordered people to remain indoors.

Cumbria police said they may never completely uncover the reason for what they described as the "most exceptional and challenging incident" the small force had ever dealt with. The alarm was raised in the harbour town of Whitehaven at 10.30am. By then, it is believed, Bird's twin brother, David, and the family solicitor, Kevin Commons, were already dead. It ended only when the gunman's body was found in a copse outside the hamlet of Boot at 1.40pm.

All 12 people killed by Bird have been named locally, but the identity of only one has so far been officially confirmed by police – that of Commons, 60, who worked for KJ Common solicitors. A statement on Cumbria police's website this morning listed 19 locations where shots were fired by Bird, including 10 sites where people were killed, and the village where he was found dead. Detectives renewed their appeal for witnesses.

"We have now concluded a 150km land and air search of the known and possible routes [Bird] could have used and as a result we are not expecting to find any further victims."

Eight people who were injured remained in hospitals in Cumbria and Newcastle upon Tyne today – four were said to be in a stable condition and four described as comfortable.

There were unconfirmed reports that the gunman, from Rowrah, near Frizington, who was divorced with two sons and had recently become a grandfather, had argued with colleagues at the taxi rank the previous night.

One friend, Peter Leder, told CNN Bird had said to him: "You won't see me again."

According to one woman in Whitehaven, Bird "shook them [his colleagues] by the hands one by one and said, 'There's going to be a rampage in this town tomorrow and it's going to start with my mother' ... They just laughed and didn't take him seriously".

Others spoke of a reported family row over the will of Bird's seriously ill mother, involving Commons. Commons's home was last night cordoned off by police, and letters from his law firm were visible on a windowsill of Bird's home.

It is thought that after killing his brother and Commons, Bird headed to Whitehaven and shot a fellow taxi driver, named locally as Darren Rewcastle. Witnesses said Bird then drove through the town with a gun hanging out of his car window. Police released a map showing Bird's progress south to Egremont, Gosforth and Seascale, where the killings continued.

Bird's spree is the biggest mass shooting since Thomas Hamilton killed 16 children and an adult at Dunblane primary school in March 1996, before killing himself.